Quadratic scattering

Suppose two gravitational waves hit a Schwarzschild black hole. Gravity is nonlinear, so the black hole responds with a third gravitational wave at the sum of their frequencies. The strength of this nonlinear response is the quadratic coupling 𝒬 we computed in Bucciotti, Redondo-Yuste, Zhong, Kuntz & Cardoso (2026). Here you can play around with the properties of the incoming and outgoing modes.

first wave
second wave
scattered wave
ℓ′

normalised by the parents’ amplitudes spheres show
Left: the two incoming waves on a sphere far away, and their wavefronts closing in. Right: the scattered wave at ω′ = ω1 + ω2, leaving. Drag a sphere to turn both. Each sphere is scaled to its own maximum, so the right one is really smaller, by ε|𝒬| for incoming waves of amplitude ε. The traces under them are the strain at the point of the sphere nearest to you.

|𝒬| over the computed (ω1, ω2) grid.
arg 𝒬, unwrapped across the grid.

Drag to turn, click to pick a point. The dot is the chosen pair; the dashed line is the resonance ω1 + ω2 = Re ωℓ′0. Grey cells touch a point whose numerical error exceeds 1% of |𝒬|.

How this is computed

The couplings were computed with BlackHoleNonlinearities.jl, which implements the calculation of the paper, for ℓ ≤ 6 and on a grid of frequencies ωi = x Re ωℓi0, x = 0.05, 0.10, …, 1.30. This page only reads that table; the sliders step through the grid, without interpolating.

The table was computed on Anvil at Purdue University through allocation PHY260445 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by U.S. National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.